kvotientrom

Kvotientrom, or quotient space, is a construction in topology that forms a new space by identifying points of a given space according to an equivalence relation. Start with a topological space X and an equivalence relation ~ on X. The elements of the quotient X/~ are the equivalence classes [x] = {y ∈ X : y ~ x}. The canonical projection q: X → X/~ sends each point x to its class [x]. The quotient topology on X/~ is defined as the finest topology for which q is continuous: a subset U ⊆ X/~ is open iff q^{-1}(U) is open in X.

Intuitively, the quotient space encodes how X is glued together by the identification specified by ~. The universal property of the quotient states that for any space Y and any continuous map f: X → Y that is constant on equivalence classes (x ~ y implies f(x) = f(y)), there exists a unique continuous map f~: X/~ → Y with f = f~ ∘ q. This characterizes X/~ as the most general space through which such maps from X factor.

Examples illustrate the idea. The real line modulo the integer relation x ~ y ⇔ x − y ∈ Z yields X/~ ≅ S^1, the circle. The interval [0,1] with its endpoints identified is homeomorphic to the unit circle. More generally, R^n modulo a lattice Z^n gives the n-torus.

Properties and caveats: The quotient can fail to be Hausdorff even when X is. The quotient of a compact space by a suitable relation remains compact under mild conditions. If the equivalence relation collapses all points to a single class, the quotient is a single point; if ~ is trivial, X/~ ≅ X.

Relation to algebraic structures: In topological groups, quotients by closed normal subgroups carry the quotient topology, forming a quotient group.